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# Lecture09A - Partition Functions for Molecules(18.3-18.8...

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Lecture 9 1 Partition Functions for Molecules (18.3-18.8) Moving beyond atoms to molecules, we have two more independent types of degrees of freedom: vibration and rotation. trans rot vib elec ε ε ε ε ε = + + + ( 29 and accordingly: q V,T trans rot vib elec q q q q = ( 29 ( 29 , and, as always: Q N,V,T ! N q V T N = First, consider a diatomic molecule: ( 29 ( 29 3/ 2 1 2 2 2 , B trans m m k T q V T V h π + = Where the total mass of the molecule (m 1 +m 2 ) is substituted for m in our monatomic solution.

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Lecture 9 2 For the electronic energy contribution for a diatomic, we need to decide what will be our zero-energy reference point. Relative to their constituent atoms, diatomics have electronic energy. Fig. 18.2 We will define separated atoms at R=∞ in their ground electronic state as our zero for energy. 2 / / 1 2 : ... e B e B D k T k T elec e e Thus q g e g e ε - = + +
Lecture 9 3 Now we can consider vibrations . v 1 v v 0,1,2... for a H.O. 2 hv ε = + = Note: in your text, the v’s (quantum numbers) look a lot like the v ’s (frequencies). Be careful! ( 29 ( 29 v v 1/2 /2 v v 0 v 0 v 0 hv hv hv vib q T e e e e β βε β β - + - - - = = = = = = 2 0 1 for 1: 1 ... 1 n n x x x x x = < = + + + = - See the back cover of your text for common expansions. ( 29 / 2 since 1, 1 hv hv vib hv e e q T e β β β - - - < = - Wow! Simple. Statistical mechanicians (mechanics?) like to simplify by using single terms (like β=1/k B T), so they define: vib B hv k Θ = vib This relates the vibrational frequency, , to a vibrational temperature, . v Θ ( 29 / 2 / 1 vib vib T vib T e q T e = -

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Lecture 9 4 2 vib ln As we noted before: E vib B d q Nk T dT = / 2 1 vib vib vib B T Nk e Θ Θ Θ = + - As I showed in lecture 6: ( 29 2 / , , 2 / 1 vib vib T V vib vib V vib T C e C R n T e Θ = = - , , 0 0 V vib V vib as T C as T C R → ∞ ( 29 ( 29 1 , 2 2 1 0.368 , 0.921 0.632 1 vib V vib e at T C R R R e - - = Θ = = = - That is, at the vibrational temperature, the heat capacity is 92.1% of the high temperature limit.
Lecture 9 5 We can also use these partition functions to calculate the population distribution as a function of temperature. Since all ideal diatomic gases will behave the same, we will consider temperatures in terms of Θ vib . ( 29 ( 29 / 2 / 2 v 1/ 2 / v 0 1 1 vib vib T hv hv vib T hv e e q T e e e β β β - - + - = = = = - - ( 29 1/ 2 th fractional population of n state.

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Lecture09A - Partition Functions for Molecules(18.3-18.8...

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